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I've been wanting to ask this question, because I have no insights into the way other mathematicians prepare papers (for eventual publication).

How much are editing, revising, updating, adding to, etc., part of the "normal" process of process of drafting math papers? Specifically, papers of moderate (10-15 pages) length. I've noticed I tend to do this for several months, and the thought has occurred to me that perhaps I'm being too "fussy" and that I'm wasting time.

My quick thoughts on the topic:

Most of the papers are not written carefully. Here are my major complains that apply to many (if not to most) of the papers that I have seen:

• Proofs are too sketchy and difficult to follow. Very often they are not correct. Even, if the mistakes are minor and easy to correct, very often the proofs are not correct the way they are written.


• Results often have complicated and abstract statements and there are no examples that would illustrate the main result.


• Authors often write something like that: `It follows from [5]' and they refer to a 500 pages long book without specifying any particular results.


• Introductions do not provide right motivation by placing the results within the existing literature.

I think it is extremely rude and unprofessional to write papers in the manner described above. This are some of the reasons why:

• Reading a paper that is not well written takes much more time both for the referee and the readers. The author perhaps saved some of his or her precious time by writing a paper not very carefully, but he or she put a burden of filling details on the shoulders of those who want to read it.


• Writing: `It follows from [5]' means that the author did not bother to check what particular result needs to be used and the reader, not familiar with [5], needs to spend hours finding the right result. Very often no result actually applies, because the result from [5] that the author had in mind, needs to be modified before it can be applied to the particular situation.

By writing a paper the author should (in my opinion):

• Keep in mind that most of the readers are graduate students who have a very limited knowledge and maturity. Papers should help them learn the subject and so the papers should have all necessary details and relevant comments placing the result in a broader framework. For example the introductions should be like a short and well written survey on the subject. At last, but not least, the papers should have no mistakes.

In my own practice I do my best to follow the rules above. Whether I am successful or not, others will judge.

• When I write a paper, I always try to write it in a way accessible to a (talented and motivated) graduate student.


• When I have a result with a complete proof and if I could write a (10 pages long) paper within a week, it usually takes me at least a month (of hard work from early morning to late night night) to write it, because of the standards that I try to follow.


• At the stage of writing a paper, I usually write about 30 pages of scrap paper for every single page in the paper. This is, because I check carefully every proof several times, each time from scratch.


• If something can be easily explained by adding a couple of lines, I do it. For example it is well known that every separable metric space can be isometrically embedded into $ell^infty$. One could quote this result from the literature, but since the proof is 1-2 lines long, why not to add such a proof?
  Of course, one needs to keep a certain balance and not add too many details. A good advice (in my opinion) is: if a paper could be written in 10 pages, it should be written on say, 13 pages.


• When I quote a result from the literature I usually state the result as a lemma (instead of saying: by Theorem 3.12 in [5]). If the statement is not exactly as in [5] I explain why the modified statement is true.

I have published over 2300 pages of technical books, and over 1000 pages of technical papers (conference and journal papers), and if there is one thing I am certain (for me, at least) it is this: writing is editing. Far far more time is spent on editing drafts than initial writing. It is inconceivable that I would or even could write any thing technical from beginning to end and not have to edit it many times.

That being said, it is true that a scholar may get caught up in finicky editing rather than more substantive scholarship and thought, so the question arises when to deem a paper "finished." Often a conference publication deadline dictates the point. Occasionally it is the knowledge that a competitor is about to "scoop" you. Sometimes it is the desire to move to a new topic or new paper that dictates the point.

My personal recommendation is to get the feedback from the most critical and knowledgeable colleague or scholar on your paper, and when he or she says it is ready for publication, go with that.

I think you should also ask yourself what your motivations are in editing and revising.

If you keep revising because you think that it is very important that published papers meet certain standards then it is an ethical choice, in a way. You will have to face consequences (publishing less) which may have an impact on your career. But there is nothing wrong in deciding that your contribution to math will come with a few nicely-written papers rather than with a huge pile of drafty ones.

There is also a possibility that you keep revising because you feel unsure about what others may think of you if you submit a less-than-perfect paper. Which is what often leads to obsessive revising, which doesn't sound as positive, does it?

In this case you should confront yourself on this point. There is a chance that after a number of published papers you'll gain some self-confidence and be more relaxed on this point; but there is also the case that at each submission your level of anxiety will increase and this will in time affect your capability of writing good math.

In any case, personally I've found that after 3-4 rounds of revision it is extremely unlikely that my paper will improve. When it happens that I change the wording of a sentence to a new wording and then I realize it's the one I started with I take it as a sign I have to stop revising.

Lastly: writing with collaborators is certainly more difficult if your writing standards are so high. But learning from others is often one of the most efficient way of learning. Give it a try!

I don't think that you should be asking about what is "normal." You should instead be asking whether your revisions are improving your paper. If your revisions are improving your paper then it's good to make them. On the other hand, there can come a point where your revisions are actually decreasing the quality of the paper. The most common cause of this is that you add things to the paper that don't really belong in the paper and actually clutter it. If you find yourself doing this then you should train yourself to resist the temptation to try to say or do too much in one paper. But if you are improving the paper then it is unproductive to worry whether what you're doing is "normal." Maybe it's not normal, but if what is normal is not good then why be normal?

Welcome to the twenty first century. "(for eventual publication)" means something different now than it did twenty years ago.

If you are building your career and have to follow certain steps to do traditional publishing in journals, then you have deadlines forced upon you (unless you have tenure, in which case your paycheck is not so deadline dependent), and that should factor into your process. I am vaguely aware of different styles my advisors used, but they were different people. I imagine both had several papers in the pipe and protocols for when to tweak and when to let go.

If you have the benefit of naming your own schedule, then I suggest ArXiv when you are not embarrassed by the resulting article, and updating every year as needed. That combines Ben McKay's "shovel" suggestion with the opportunity of "error tracking" as suggested in RBega's comment above. I think this method is very effective if you have a large goal in mind, such as rearranging your work in book form for students to follow. (Replace every year by your own interval, but be careful about overly frequent public changes.)

Gerhard "You Worry About Version Tracking" Paseman, 2018.09.27.

Unless you have a need to get the paper published asap, I recommend you put it aside for a month or so instead of looking through it repeatedly. When you next read the paper, many of your preconceptions and assumptions will have been forgotten and you will be much more likely to notice things like missing definitions, logical gaps, and sections of proofs that are going to confuse readers.